60.7.91 problem 1682 (book 6.91)

Internal problem ID [11680]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1682 (book 6.91)
Date solved : Tuesday, January 28, 2025 at 06:07:24 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 9 x^{2} y^{\prime \prime }+a y^{3}+2 y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 35

dsolve(9*x^2*diff(diff(y(x),x),x)+a*y(x)^3+2*y(x)=0,y(x), singsol=all)
 

\[ y = c_{2} \operatorname {JacobiSN}\left (\frac {\left (2 c_{1} x^{3}+\sqrt {2}\, \sqrt {x^{{20}/{3}} a}\right ) c_{2}}{2 x^{3}}, i\right ) x^{{1}/{3}} \]

✓ Solution by Mathematica

Time used: 3.395 (sec). Leaf size: 41

DSolve[2*y[x] + a*y[x]^3 + 9*x^2*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 \sqrt [3]{x} \text {sn}\left (\left .\left (c_1+\frac {\sqrt {a x^{20/3}}}{\sqrt {2} x^3}\right ) c_2\right |-1\right ) \]